Convert the following repeating decimal to a fraction in simplest form. \[ .1 \overline{8} \]

To convert the repeating decimal \(0.1\overline{8}\) to a fraction, let \(x = 0.1\overline{8}\). Then we can express this as: \[ x = 0.188888\ldots \] Multiply both sides by 10 to get rid of the decimal point in the tenths place: \[ 10x = 1.88888\ldots \] Next, notice that if we multiply both sides by 100 (to shift the repeating decimal two places), we can write: \[ 1000x = 188.8888\ldots \] Now we have two equations: 1. \(10x = 1.88888\ldots\) (Equation 1) 2. \(1000x = 188.8888\ldots\) (Equation 2) Now, if we subtract Equation 1 from Equation 2, we get: \[ 1000x - 10x = 188.8888\ldots - 1.88888\ldots \] This simplifies to: \[ 990x = 187 \] Next, solve for \(x\): \[ x = \frac{187}{990} \] Now we need to check if this fraction can be simplified. The greatest common divisor of 187 and 990 is 1, so the fraction is already in simplest form. Thus, the repeating decimal \(0.1\overline{8}\) as a fraction in simplest form is: \[ \frac{187}{990} \]